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Full text of "Treatise On Analysis Vol-Ii"

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1.   (a)   Let G be a topological group, K its identity component, H a subgroup of G
contained in K. Show that the connected components of the space G/H are the images
of the connected components of G under the canonical mapping TT : G --G/H. Show
that K is the smallest of the subgroups L of G such that G/L is totally disconnected.

(b)    Let G be the additive subgroup of the Banach space <^R(N) (7.1.3) consisting of
the mappings n\-*f(ri) such that/(tf) e Q for all n e N and lim /(/?) exists in R. The

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distance induced on G by that on ^R(N) defines a topology compatible with the group
structure of G, and with respect to this topology G is totally disconnected. Let H be the
subgroup of G consisting of all /e G such that lim f(n) = 0. Show that H is closed


in G and that G/H is isomorphic to R, and therefore connected.

(c)   Let G be a locally compact metrizable group, H a closed subgroup of G, and
TT : G ~> G/H the canonical mapping. Show that the connected components of G/H
are the closures of the images under TT of the connected components of G. (Reduce
to the case where G is totally disconnected, and use Problem 3 of Section 12.9.)

2.   In the additive group R, let H be the subgroup Z and let A be the subgroup 0Z, where
6 is an irrational number. Show that the canonical bijection A/(A n H) -> (A + H)/H
is not an isomorphism of topological groups.

3.    Let/? be a prime number and let (Gn) be an infinite sequence of topological groups all
equal to the discrete group Z/p2Z. Let Hn be the subgroup pZ/p2Z of Gn. Let G be the
subgroup of the product group fj Gn consisting of all x = (xn) such that xn e Hn for all


but a finite set of values of n. Let S3 be the set of neighborhoods of 0 in the product
group fj Hn = H c G. Show that S3 is a fundamental system of neighborhoods of 0


for a topology on G compatible with the group structure, and that in this topology G
is metrizable and locally compact and G/H is discrete. Show that the homomorphism
u: x\-+px of G into G is not a strict rnorphism of G into G, and that u(G) is not closed

4.    Let G be a metrizable group, K a closed normal subgroup of G. If K and G/K are
complete, show that G is complete.

5.   (a)   Let G be a topological group, K a normal subgroup of G. Show that if K and
G/K have no small subgroups (Section 12.9, Problem 6), then G has no small subgroups,
(b)   Deduce from (a) that if HI, H2 are two normal subgroups of a topological group
G, such that G/H! and G/H2 have no small subgroups, then G/(Hi n H2) has no small

6.    Let G be a connected, locally compact, metrizable group, K a compact normal sub-
group of G, and N a closed normal subgroup of K, such that K/N has no small sub-
groups (Section 12.9, Problem 6). Show that N is a normal subgroup of G. (Observe
that the hypothesis on K/N implies the existence of a neighborhood U of e in G such
that xNx"1 = N for all x e U.)

7.    Let G be a connected metrizable group and H a compact normal commutative sub-
group of G, with no small subgroups. Show that H is contained in the center of G.0. Now use the compactness of E to complete the proof.)